How the Square Root Calculator Works
A square root answers the question: which number, multiplied by itself, gives this number? The square root of 144 is 12, because 12 x 12 = 144. This calculator also finds cube roots and any other root, such as the 4th root or the 5th root. It is useful for geometry, algebra, science and everyday measuring.
What you enter
- Radicand: the number you want the root of (the default is 144).
- Root degree: which root to find. 2 is the square root, 3 is the cube root, 4 is the 4th root, and so on (the default is 2).
The formula
nth root of x = x ^ (1 / n)
A root is the same as a power with a fraction. The square root of x is x to the power of 1/2, and the cube root is x to the power of 1/3. The answer is rounded to 6 decimal places.
A worked example
| Question | Answer |
|---|---|
| Square root of 144 | 12 |
| Square root of 2 | 1.414214 |
| Cube root of 27 | 3 |
| Cube root of -8 | -2 |
| 4th root of 81 | 3 |
| 5th root of 32 | 2 |
Perfect squares
A perfect square is a whole number whose square root is also a whole number:
| Number | Square root |
|---|---|
| 1 | 1 |
| 4 | 2 |
| 9 | 3 |
| 16 | 4 |
| 25 | 5 |
| 36 | 6 |
| 100 | 10 |
| 144 | 12 |
Chart: square roots of perfect squares
The numbers grow fast, but the roots grow slowly. The bar is the size of the root:
root of 1 ███████ 1 root of 4 █████████████ 2 root of 9 ████████████████████ 3 root of 16 ███████████████████████████ 4 root of 25 █████████████████████████████████ 5 root of 36 ████████████████████████████████████████ 6
Chart: different roots of 64
The higher the degree, the smaller the answer (for numbers above 1):
square root (2) ████████████████████████████████████████ 8 cube root (3) ████████████████████ 4 6th root (6) ██████████ 2
Negative numbers
- Odd roots work: the cube root of -8 is -2, because (-2) x (-2) x (-2) = -8.
- Even roots do not: the square root of -4 has no real answer (it is imaginary), so the calculator shows a message.
- A degree of 0 is not allowed, because it would mean dividing by zero.
Doing it by hand
- Find the two perfect squares around your number. The square root of 50 is between 7 (49) and 8 (64), and it is just over 7. The exact answer is 7.071068.
- To check an answer, multiply it by itself. 12 x 12 = 144.
- The square root of a product is the product of the roots: the square root of 36 x 4 is 6 x 2 = 12.
Where roots are used
- Geometry: the side of a square from its area, or the long side of a right triangle.
- Science and finance: standard deviation and compound growth rates use roots.
- Home projects: the length of a diagonal.
This tool gives an estimate for general information only. Answers are rounded to 6 decimal places.